Era III — The Golden Age (c. 1000–1100)¶
The eleventh century produced several of the greatest scientists of the entire medieval world, East or West: Ibn al-Haytham, al-Biruni, and (spilling into the twelfth century) Omar Khayyam. Meanwhile al-Andalus matured into a mathematical center whose output would soon flow into Latin.
Abu Nasr Mansur¶
Abu Nasr Mansur ibn Ali ibn Iraq (c. 960 – c. 1036) · Khwarazm, Ghazna · trigonometry, geometry
A prince of the Khwarazmian ruling house, teacher and lifelong collaborator of al-Biruni (they exchanged dedicated treatises for decades). One of the three claimants to the spherical law of sines, and author of the definitive Arabic edition of Menelaus's Spherics, the foundation text of spherical geometry, which survives through his recension. His work systematically replaced the cumbersome Menelaus configuration theorem with the cleaner sine-rule methods that made spherical astronomy computationally humane.
Sources: DSB; Berggren, Episodes, ch. 6; Kennedy.
al-Nasawi¶
Abu al-Hasan Ali ibn Ahmad al-Nasawi (c. 1011 – c. 1075) · Khurasan, Baghdad · arithmetic
Student of Kushyar ibn Labban's tradition; his al-Muqni fi al-Hisab al-Hindi (written first in Persian, then Arabic) continued the decimal-arithmetic textbook line, treating fractions, root extraction (with the approximation refinements typical of the school), and criticizing his predecessors' expositions. Useful to historians precisely because he reviews and compares earlier methods.
Sources: DSB; Saidan's studies of Arabic arithmetic.
Ibn Tahir al-Baghdadi¶
Abd al-Qahir ibn Tahir al-Baghdadi (d. 1037) · Baghdad, Nishapur · arithmetic, number theory
Theologian and mathematician; his al-Takmila fi al-Hisab ("Completion of Arithmetic") surveys the number systems in concurrent use, Indian decimal, finger-reckoning, sexagesimal, and treats topics in number theory including a result on equivalent numbers (pairs with equal sums of proper divisors of some value) beyond the Greek material, and an improved discussion of amicable and perfect numbers.
Sources: Saidan's edition and studies; DSB.
Ibn al-Haytham¶
Abu Ali al-Hasan ibn al-Haytham (c. 965 – c. 1040) · Basra, then Cairo · optics, geometry, astronomy, method
Latinized as Alhazen (from al-Hasan). One of the most important scientists of the Middle Ages in any civilization.
Optics. The seven-book Kitab al-Manazir ("Book of Optics," c. 1011–1021, composed in Cairo) refounded the science of vision on the intromission theory (light travels from objects to the eye), fused the mathematical (Euclid, Ptolemy) and medical (Galen) traditions, and, decisively for the history of science, argued by systematic experiment: controlled setups with dark chambers (the camera obscura analyzed properly for the first time), apertures, and measured refractions. "Alhazen's problem", find the reflection point on a spherical mirror between given object and eye, reduces to a quartic he solved geometrically via conics; algebraic solutions were only achieved by European mathematicians centuries later (Huygens gave a famous treatment).
Mathematics. He summed the fourth powers \(\sum k^4\) (and gave a method extending to any integer power) using a telescoping/recursive argument, and used the result to compute the volume of a paraboloid of revolution about its base, a genuine integration in all but name; the summation formulas are exactly the ones Cavalieri and Fermat needed for \(\int x^n\,dx\) in the seventeenth century. He attempted the parallel postulate via what is now called the "Lambert quadrilateral" configuration (three right angles, investigate the fourth), studied lunes and the quadrature of circle-related figures, and wrote On the Analysis and Synthesis of Mathematical Problems.
Astronomy. His Doubts Concerning Ptolemy (al-Shukuk ala Batlamyus) catalogued the physical inconsistencies of Ptolemaic planetary models (equant, prosneusis), setting the critical agenda that the Maragha school later executed.
Transmission. The Optics was translated into Latin around 1200 as De aspectibus / Perspectiva and shaped everyone: Roger Bacon, Witelo, Pecham, the Renaissance perspectivists (Ghiberti cited him), and Kepler, whose Ad Vitellionem Paralipomena (1604), the founding work of modern optics, is framed as a supplement to the Alhazen-Witelo tradition. Risner's 1572 printed edition Opticae thesaurus was standard into the seventeenth century.
Sources: A. I. Sabra's edition and English translation of the Optics, Books I–III (1989); Sabra's DSB article "Ibn al-Haytham"; Rashed, Les mathématiques infinitésimales du IXe au XIe siècle; Katz; A. Mark Smith, From Sight to Light (2015).
al-Biruni¶
Abu Rayhan Muhammad ibn Ahmad al-Biruni (973–1048) · Khwarazm, Ghazna · mathematics, astronomy, geodesy, history of science
Perhaps the most universal scholar of the era: mathematician, astronomer, geographer, historian, linguist (he learned Sanskrit and wrote the great ethnographic-scientific study India), and mineralogist. Mathematical highlights:
Geodesy. His measurement of the Earth's radius by observing the dip of the horizon from a mountain of measured height (in the Punjab region) is a famous set piece of applied trigonometry, requiring only an astrolabe and the mountain, and yielding a value close to modern figures (manuscript unit uncertainties make exact accuracy claims slippery; the method is the point).
Trigonometry. His Qanun al-Mas'udi (the "Mas'udic Canon," c. 1030s), an astronomical encyclopedia, contains state-of-the-art trigonometric tables and methods; his separate treatise on chords collects and proves a wealth of circle geometry. He treated the qibla problem definitively in The Determination of the Coordinates of Cities, a monograph of mathematical geography.
Numerical analysis. To trisect angles / compute \(\sin 1°\) (which leads to an irreducible cubic), he used iterative and interpolation techniques of high sophistication, and he explicitly discussed the cubic equation for the regular nonagon's chord.
The shadow. His Exhaustive Treatise on Shadows is the fullest medieval study of the tangent and cotangent functions and gnomonics.
Transmission. Little of al-Biruni was translated into Latin, making him the standing example that the Islamic tradition's ceiling was far above what Europe received. His work was hugely influential eastward (Persia, India) and is now a primary source for historians of multiple civilizations.
Sources: E. S. Kennedy's numerous al-Biruni studies; DSB (Kennedy, "al-Biruni"); Berggren, Episodes, ch. 6; English translations: Sachau (India, Chronology), Jamil Ali (Coordinates of Cities), Kennedy (Shadows).
Ibn Sina¶
Abu Ali al-Husayn ibn Sina (980–1037) · Bukhara, Persia · philosophy, medicine, with mathematical sections
Latinized as Avicenna. Included here for completeness: his encyclopedic Kitab al-Shifa ("Book of Healing") contains substantial mathematical sections, a recension of Euclid, an arithmetic treating topics like the casting-out-of-nines check with a proof-oriented attitude, music theory, and astronomy, which shaped how mathematics was taught within philosophy curricula for centuries in both the Islamic world and (via Latin translation of parts of the Shifa) Europe. Not an original mathematician of the first rank, but a major channel of mathematical culture.
Sources: DSB; Katz.
Ibn al-Samh and the school of al-Andalus¶
Ibn al-Samh (984–1035, Granada) · geometry, astronomy · and al-Zarqali (Azarquiel) (c. 1029–1100, Toledo/Córdoba) · astronomy, instruments
Ibn al-Samh, a student of Maslama al-Majriti's school, wrote on the cylinder's sections (his treatise on the ellipse as a plane section survives through Hebrew transmission) and on the astrolabe. Al-Zarqali, a generation later, is the towering Andalusi figure: principal author of the Toledan Tables, the zij (adapted from al-Khwarizmi, al-Battani, and new Toledo observations) that, in Latin translation, became the standard astronomical tables of Europe for two centuries, until superseded by the Alfonsine Tables (which were themselves produced by a Castilian court program working in the Arabic tradition). He invented the safiha (saphaea), a universal astrolabe plate usable at any latitude, described Mercury's orbit with an oval curve, and his value for the motion of the solar apogee was cited by Copernicus in De revolutionibus.
Sources: DSB (J. Vernet, "al-Zarqali"); Samsó; Kennedy, Survey; Toomer's study of the Toledan Tables.
Ibn Muadh al-Jayyani¶
Abu Abdallah Muhammad ibn Muadh al-Jayyani (989 – c. 1079) · Jaén, al-Andalus · trigonometry, astronomy
Author of The Book of Unknown Arcs of a Sphere, the first treatise devoted to spherical trigonometry as an independent subject (rather than a chapter of astronomy), systematically solving spherical triangles from given parts. Whether he drew on the eastern school (Abu Nasr et al.) or worked independently is debated; either way the treatise marks trigonometry's arrival in al-Andalus as a mature discipline, conveniently placed for transmission northward. His treatise on twilight (estimating atmospheric refraction height) circulated in Latin, long misattributed to Alhazen.
Sources: M. V. Villuendas's edition; DSB; Van Brummelen, The Mathematics of the Heavens and the Earth.
al-Mu'taman ibn Hud¶
Yusuf al-Mu'taman ibn Hud (d. 1085) · Zaragoza · geometry
King of the taifa of Zaragoza (r. 1081–1085) and a serious research geometer, a combination with few parallels. His Kitab al-Istikmal ("Book of Perfection"), rediscovered and analyzed by Jan Hogendijk in the 1980s, is a synthesis of the Greek and Arabic geometric corpus containing original results, including the theorem on the ratios cut on a triangle's cevians now known as Ceva's theorem, six centuries before Giovanni Ceva (1678). No transmission route from Zaragoza to Ceva is documented; it stands as an independent anticipation. The Istikmal circulated east (Maimonides knew and taught it).
Sources: J. P. Hogendijk, "Discovery of an 11th-century geometrical compilation: the Istikmal of Yusuf al-Mu'taman ibn Hud," Historia Mathematica 13 (1986); DSB supplement.
Omar Khayyam¶
Ghiyath al-Din Abu al-Fath Umar ibn Ibrahim al-Khayyami (1048–1131) · Nishapur, Isfahan · algebra, geometry, astronomy (and, by attribution, poetry)
In the West famous for the Rubaiyat (the attribution of much of that poetry is a separate scholarly tangle); in the history of mathematics, famous for two achievements.
The cubics. His Treatise on Demonstration of Problems of Algebra (c. 1070, Isfahan) gives the first systematic theory of cubic equations: he classifies all cubics (fourteen types not reducible to lower degree, under positive-coefficient conventions), and solves every type geometrically by intersecting conic sections, a circle with a parabola, two hyperbolas, and so on, proving the constructions correct. He states frankly that he could not find arithmetical (algebraic) solutions and expresses hope that successors would, a hope fulfilled in 1520s–1540s Italy (del Ferro, Tartaglia, Cardano). He also recognized that cubics can have two (positive) roots, and his remark that "algebra is not a collection of tricks but geometric facts proved" captures the discipline's new self-understanding. Whether Italian algebraists knew Khayyam's text is doubtful (no Latin translation is known); the geometric-conics approach, however, reappears in Descartes' Géométrie (1637), which solves cubics and quartics by intersecting conics in exactly Khayyam's spirit.
The parallels and ratios. His Explanation of the Difficulties in the Postulates of Euclid (1077) attacks the parallel postulate via the birectangular quadrilateral (equal sides perpendicular to the base; decide whether the summit angles are right, acute, or obtuse), the exact figure later called the Saccheri quadrilateral (1733), and refutes the acute/obtuse cases only by importing an equivalent postulate, as all such attempts must. The same work deepens the theory of ratio and proportion, arguing for treating ratios as numbers, a step toward the modern real-number concept.
The calendar. As head of the Isfahan observatory under Malik-Shah, he led the reform producing the Jalali calendar (1079), whose intercalation scheme yields a mean year accurate to roughly a day in several thousand years, more accurate than the Gregorian rule.
Sources: Rashed & Vahabzadeh, Omar Khayyam the Mathematician (2000); Berggren, Episodes, ch. 4; Katz; DSB (Youschkevitch & Rosenfeld, "al-Khayyami"); Rosenfeld, A History of Non-Euclidean Geometry.